Problem: Multiply the following complex numbers, marked as blue dots on the graph: $(5 e^{19\pi i / 12}) \cdot (2 e^{\pi i / 2})$ (Your current answer will be plotted in orange.)
Answer: Multiplying complex numbers in polar forms can be done by multiplying the lengths and adding the angles. The first number ( $5 e^{19\pi i / 12}$ ) has angle $\frac{19}{12}\pi$ and radius $5$ The second number ( $2 e^{\pi i / 2}$ ) has angle $\frac{1}{2}\pi$ and radius $2$ The radius of the result will be $5 \cdot 2$ , which is $10$ The sum of the angles is $\frac{19}{12}\pi + \frac{1}{2}\pi = \frac{25}{12}\pi$ The angle $\frac{25}{12}\pi$ is more than $2 \pi$ . A complex number goes a full circle if its angle is increased by $2 \pi$ , so it goes back to itself. Because of that, angles of complex numbers are convenient to keep between $0$ and $2 \pi$ $\frac{25}{12}\pi - 2 \pi = \frac{1}{12}\pi$ The radius of the result is $10$ and the angle of the result is $\frac{1}{12}\pi$.